00:01
This problem says find the absolute maximum and absolute minimum values of f on the given interval.
00:05
And we're given the function f of x equals 6x cubed minus 18x squared minus 54x plus 5 on the interval negative 2 to 4 with negative 2 and 4 included.
00:15
And to find the absolute minimum and absolute maximum value, what we can do is find the potential locations of extrema on our function by looking at the first derivative of f of x.
00:25
And to do that, we will bring down our 3 to multiply by 6 to give us 18x to the 3 minus 1 to the second power.
00:32
2 will multiply by negative 18 to give us negative 36x to the 2 minus 1 to the first power.
00:37
The derivative of any linear term is just the coefficient, and the derivative of any constant is just 0.
00:42
So we will then set this equal to 0 again to solve for the potential x values or inputs of these local maximums and local minimums so we can compare to see what's the absolute max and absolute min.
00:53
And to solve here, we'll factor out 18, which leaves us with x squared minus 2x and then minus 3 equal to 0.
01:01
Now we want the two numbers to multiply to be negative 3 and add to be negative 2, which would be negative 3 and positive 1, still with our 18 in front, equal to 0...